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A195527 Integers n that are k-gonal for precisely 3 distinct values of k, where k >= 3.

%I A195527 #12 Jul 28 2016 21:15:51
%S A195527 15,21,28,51,55,64,70,75,78,91,96,100,111,112,117,126,135,136,141,144,
%T A195527 145,148,154,156,165,175,176,186,189,195,201,204,216,232,235,238,246,
%U A195527 255,256,285,286,288,291,297,300,306,315,316,321,322,324,330,333,336
%N A195527 Integers n that are k-gonal for precisely 3 distinct values of k, where k >= 3.
%C A195527 See A177025 for number of ways a number can be represented as a polygonal number.
%H A195527 Chai Wah Wu, <a href="/A195527/b195527.txt">Table of n, a(n) for n = 1..10000</a>
%e A195527 21 is in the sequence because it is a triangular number (A000217), an octagonal number (A000567) and an icosihenagonal number (A051873).
%t A195527 data1=Reduce[1/2 n (n(k-2)+4-k)== # && k>=3 && n>0, {k,n}, Integers]&/@Range[336]; data2=If[Head[#]===And, 1, Length[#]] &/@data1; data3=DeleteCases[Table[If[data2[[k]]==3, k], {k, 1, Length[data2]}], Null]
%o A195527 (Python)
%o A195527 A195527_list = []
%o A195527 for m in range(1,10**4):
%o A195527     n, c = 3, 0
%o A195527     while n*(n+1) <= 2*m:
%o A195527         if not 2*(n*(n-2) + m) % (n*(n - 1)):
%o A195527             c += 1
%o A195527             if c > 2:
%o A195527                 break
%o A195527         n += 1
%o A195527     if c == 2:
%o A195527         A195527_list.append(m) # _Chai Wah Wu_, Jul 28 2016
%Y A195527 Cf. A000217, A000567, A051873, A177025, A177029.
%K A195527 nonn
%O A195527 1,1
%A A195527 _Ant King_, Sep 21 2011